Binomials Arising from Buchberger Algorithm on Polyomino Ideals

A polyomino is a finite set of unit squares joined side by side on the Cartesian plane. Qureshi introduced an ideal constructed from a polyomino which is called "polyomino ideal". In this paper, we study the binomials arising from Buchberger Algorithm on polyomino ideals. We also introduce socket wrench polyominoes and study the Gr"{o}bner bases of the ideal $I_{\mathcal P}$ and some algebraic properties of $K[\mathcal P]$.